On scrambled sets and a theorem

of Kuratowski on independent sets

Author:
Hisao Kato

Journal:
Proc. Amer. Math. Soc. **126** (1998), 2151-2157

MSC (1991):
Primary 54H20, 26A18

DOI:
https://doi.org/10.1090/S0002-9939-98-04344-5

MathSciNet review:
1451813

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Abstract: The measure of scrambled sets of interval self-maps was studied by many authors, including Smítal, Misiurewicz, Bruckner and Hu, and Xiong and Yang. In this note, first we introduce the notion of ``-chaos" which is related to chaos in the sense of Li-Yorke, and we prove a general theorem which is an improvement of a theorem of Kuratowski on independent sets. Second, we apply the result to scrambled sets of higher dimensional cases. In particular, we show that if a map of the unit -cube is -chaotic on , then for any there is a map such that and are topologically conjugate, and has a scrambled set which has Lebesgue measure 1, and hence if , then there is a homeomorphism with a scrambled set satisfying that is an -set in and has Lebesgue measure 1.

**1.**B A. M. Bruckner and T. Hu, On scrambled sets for chaotic functions, Trans. Amer. Math. Soc. 301 (1987), 289-297. MR**88f:26003****2.**W. J. Gorman III, The homeomorphic transformation of c-sets into d-sets, Proc. Amer. Math. Soc. 17 (1966), 825-830. MR**34:7734****3.**H. Kato, Attractors and everywhere chaotic homeomorphisms in the sense of Li-Yorke on manifolds and -dimensional Menger manifolds, Topology Appl. 72 (1996), 1-17.**4.**Ku K. Kuratowski, Applications of Baire-category method to the problem of independent sets, Fund. Math. 81 (1973), 65-72. MR**49:3855****5.**K. Kuratowski, Topology I, PWN, Warsaw, 1966. MR**36:840****6.**T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. MR**52:5898****7.**M. Misiurewicz, Chaos almost everywhere, Lecture Notes in Math. 1163 (1985), 125-130. MR**87e:58152****8.**E. E. Moise, Geometric topology in dimension 2 and 3, Springer, 1977. MR**58:7631****9.**J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. 42 (1941), 873-920. MR**3:211b****10.**T. B. Rushing, Topological embeddings, Academic Press, New York, 1973. MR**50:1247****11.**J. Smítal, A chaotic function with some extremal properties, Proc. Amer. Math. Soc. 87 (1983), 54-56. MR**84h:26008****12.**J. Smítal, A chaotic function with a scrambled set of positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), 50-54. MR**86b:26009b****13.**J. Xiong and Z. Yang, Chaos caused by a topologically mixing map, World Scientific, Advanced Series in Dynamical Systems Vol 9, 550-572. MR**93c:58153**

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Additional Information

**Hisao Kato**

Affiliation:
Institute of Mathematics, University of Tsukuba, Ibaraki, 305 Japan

Email:
hisakato@sakura.cc.tsukuba.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-98-04344-5

Keywords:
Scrambled set,
independent set,
Cantor set,
flat,
Lebesgue measure

Received by editor(s):
August 29, 1996

Received by editor(s) in revised form:
December 20, 1996

Communicated by:
Mary Rees

Article copyright:
© Copyright 1998
American Mathematical Society